Theorem csbiebt 3867

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3871.) (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbiebt	⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebt

Step	Hyp	Ref	Expression
1		elex 3453	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		spsbc 3743	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	adantr 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → 𝐴 ∈ V)
5		biimt 361	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
6		csbeq1a 3852	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	eqeq1d 2742	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 7	bitr3d 282	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	8	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
10		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ V
11		nfnfc1 2905	. . . . . 6 ⊢ Ⅎ𝑥Ⅎ𝑥𝐶
12	10, 11	nfan 1906	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ Ⅎ𝑥𝐶)
13		nfcsb1v 3862	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
15		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥𝐶)
16	14, 15	nfeqd 2912	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
17	4, 9, 12, 16	sbciedf 3772	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
18	3, 17	sylibd 240	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
19	13	a1i 11	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
20		id 22	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥𝐶)
21	19, 20	nfeqd 2912	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
22	11, 21	nfan1 2212	. . . . . 6 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
23	7	biimprcd 251	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
24	23	adantl 482	. . . . . 6 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → (𝑥 = 𝐴 → 𝐵 = 𝐶))
25	22, 24	alrimi 2225	. . . . 5 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
26	25	ex 413	. . . 4 ⊢ (Ⅎ𝑥𝐶 → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
27	26	adantl 482	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
28	18, 27	impbid 213	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
29	1, 28	sylan 586	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  Ⅎwnfc 2887  Vcvv 3432  [wsbc 3730  ⦋csb 3838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-v 3434  df-sbc 3731  df-csb 3839

This theorem is referenced by:  csbiedf  3868  csbieb  3869  csbiegf  3871